This may be taken as an alternative definition of sheaf: since Lawvere-Tierney topologies are bijectively given by geometric embeddings, instead of explicitly defining a sheaf as a presheaf satisfying descent, one may define categories of sheaves as geometric embeddings into presheaf categories.

Giraud's axiomatic characterization

Giraud characterized Grothendieck toposes as categories satisfying certain exactness and smallcompleteness properties (where “small” is again relative to the given category of sets SS). The exactness properties are elementary (not depending on SS), and are satisfied in any elementary topos, or even a pretopos.

See the Elephant, theorem C.2.2.8. (There, the assumption of local smallness is not stated explicitly, but it is included in the definition of ∞\infty-pretopos by way of well-poweredness; on the nLab it is not so included, so we have to state it explicitly. To see that it is necessary, note that if UU and VV are Grothendieck universes with U∈VU\in V, then SetVSet_V satisfies all the other conditions relative to SetUSet_U, but is not locally small and is not a Grothendieck topos.) See also Wikipedia.

Sometimes (3,4) are combined and strengthened to the statement that the category has all small colimits, which are effective and pullback-stable. However, this is a mistake for two reasons: it is a significantly stronger axiomatisation (since without the small generating set, not every infinitary pretopos has this property), and it is not valid in weak foundations (while the definition given above is).

Street’s axiomatic characterization

Augmenting the aforementioned Proposition that Grothendieck toposes are total categories, Street more or less characterizes Grothendieck toposes as lex total categories having the same “size” as SetSet.

In more detail: we take as background set theory ZFC + “there is a strongly inaccessible cardinal” κ\kappa; equivalently, the existence of a single Grothendieck universeUU (a set of “small sets”). Supposing given a model VV of ZFC+universeZFC+universe, “category” shall then refer to category theory interpreted in VV. Let SetSet be the category of small sets; note that the set of morphisms of SetSet has size κ\kappa.

Theorem

(Street) A category EE is a Grothendieck topos iff it is lex total and the size of the set of isomorphism classes of objects is κ\kappa or less.

Remark

This result is in the spirit of saying “every Grothendieck topos is the category of sheaves with respect to the canonical topology on itself”. Putting aside set-theoretic issues, it suggests that Grothendieck toposes be seen as analogous to frames, which may be defined as lex total objects in 2\mathbf{2}-CatCat. In this setting, the appropriate morphisms are left exact left adjoints, so that Grothendieck toposes and geometric morphisms between them would be analogous to locales and continuous maps between them.

In weak foundations

We have two definitions of a Grothendieck topos:

the category of sheaves on some small site,

a category that satisfies Giraud's axioms (as listed above).

The theorem that these are equivalent can be proved in quite weak foundations, whether finitist, predicative, or constructive (or all three at once), as long as we axiomatize correctly given the caveats listed in the previous section. Some hard-nosed predicativists (and even hard-nosed ZFC fundamentalists) may object to the language (on the ground that large categories such as SetSet and other nontrivial Grothendieck toposes don't really exist), but they should accept the theorems when suitably phrased.

In predicative mathematics, however, we cannot prove that every Grothendieck topos is in fact a topos! In fact, it is immediate that the category of sets is a Grothendieck topos, but SetSet is an elementary topos if and only if power sets are small, which is precisely what predicativists doubt. One can use the term Grothendieck pretopos to avoid implying that we have an elementary topos. On the other hand, since Grothendeick toposes came first, perhaps it is the definition of ‘elementary topos’ that is too strong.

Similarly, in finitist mathematics, we cannot prove that every Grothendieck topos has a natural numbers object; while in strongly predicative mathematics, we cannot prove that every Grothendieck topos is cartesian closed. In each case, once a property is accepted of SetSet (the axiom of infinity and small function sets, in these examples), it can be proved for all Grothendieck toposes.

Constructivism as such is irrelevant; even in classical mathematics, most Grothendieck toposes are not boolean. However, for an analogous result, try the theorem that the category of presheaves on a groupoid (and hence any category of sheaves contained within it) is boolean. (Again, SetSet itself is an example of this.)

The theorem that every Grothendieck topos is cocomplete is a subtle point; it fails only in finitist predicative mathematics. (The key point in the proof is to generate the transitive closure∼*\sim^* of a binary relation ∼\sim. One proof defines a∼*ba \sim^* b to mean that a∼x0∼⋯∼xn−1∼ba \sim x_0 \sim \cdots \sim x_{n-1} \sim b for some nn, which is predicative but infinitary; another defines a∼*ba \sim^* b to mean that a∼′ba \sim' b for every transitive relation ∼′\sim' that contains ∼\sim, which is finitary but impredicative.)

History

At the start of 1958 Grothendieck believed the correct Weil cohomology of a schemeSS would be the derived functor cohomology of some categoryAb(S)Ab(S) of sheaves of Abelian groups on SS —and Ab(S)Ab(S) was likely to be the category of all Abelian group objects in some category of sheaves of sets on SS. He had no name yet for such conjectural categories of sheaves of sets but for later reference I will call this category T(S)T(S). It would require some new notion of sheaves of sets more general than the existing notion using topological spaces. He had no concrete idea what this new notion of sheaf might be. His experience in Kansas in 1955 suggested it was likely to come from some new notion of espace etale over a scheme. and it should have some good exactness properties.

On April 21, 1958 Grothendieck went to the Seminaire Chevalley to hear Serre describe isotrivial fiber bundles on a variety V, which are bundles that become trivial when pulled back to some surjective family of finite, unramified maps to V. Serre showed this gave the expected one-dimensional Weil cohomology groups of V. Grothendieck immediately told Serre it would work in all dimensions–which Serre found “very optimistic”“ at the time.

All of this is well documented in familiar sources.

Grothendieck’s 1973 topos lectures in Buffalo show that during Serre’s talk Grothendieck saw the Weil espaces etales over a scheme SS should be patched together from Serre’s unramified maps to SS. That is, the sought category T(S)T(S) of sheaves of sets on S should be generated by the unramified maps to S, and should have arbitrary colimits of these, and finite limits. He first sought to axiomatize this roughly the way his Tohoku paper had axiomatized sheaves of groups.

A few changes came soon: He realized finite unramified maps should be replaced by flat unramified maps (just as topological espaces etales are trivial locally in the fiber and not on the base). He abandoned the axiomatic approach to T(S)T(S) as too vague and shifted to construction by sites. And for sites he began treating sheaves as a kind of functor, rather than as espaces etales. Only after the notion ot site developed would Giraud give his topos axioms, and no one has yet really taken a precise notion of espace etale much beyond the topological case.

But April 21 1958 was the birth of topos theory. The term topos came later. (Lots of people are not named on the day they are born.)

As of the summer of 1973 Grothendieck’s stated preferred definition of topos was still: a category with arbitrary colimits, finite limits, and a small generating set. He says over and over this is not quite adequate for proofs. He says proofs require the notion of site, or else the Giraud axioms, but he calls the vaguer idea more intuitive and says that is the way to think about a topos.

Also in 1973 Grothendieck says the objects in any topos should be seen as espaces etales over the terminal object of the topos, in a generalized sense that includes saying any orbit of a group action lies “etale” over a fixed point. Today, it is not obvious that this can work well in general. In SGA4 Grothendieck had already given evidence that it would work for petit topos but not gros topos. But still in 1973 he did say it. And it works perfectly for petit etale toposes—as long as we generalize the notion of espace etale to include a Galois orbit (in an extension field) lying over a single point (in the ground field).

References

A quick introduction of the basic facts of sheaf-topos theory is chapter I, “Background in topos theory” in